Getting from the Texas Hill Country to the Dalmatian Coast in a lie-flat seat requires strategy, since no carrier offers a one-stop shuttle worth bragging about—there is no nonstop, and booking business class from Austin to Split nearly always means routing through a major East Coast or European hub with a connection onward, typically via Charlotte, JFK, or a European gateway before a short intra-European hop into Split's compact airport. American, Delta, and United all serve this corridor with codeshare and joint-venture partners across the Atlantic, and the smart move is pairing AA or Delta's domestic long-haul metal—think Delta's A330 or A350 business class with lie-flat suites, or American's Flagship Business on their 777s and 787s—with a European partner for the Split leg, since none of the big three fly their own metal directly into SPU.
Having routed through this itinerary more than a few times, I've learned that Split's airport gets overwhelmed quickly in peak summer, so if your connection lands you there between June and August, budget extra time for a single, slow-moving passport hall—arriving via the shoulder season sidesteps this entirely. That's also when the fares make sense: April, May, September, and October not only avoid the Adriatic crowds but consistently produce the best premium-cabin availability, since airlines open up more saver business class awards and revenue fares once the leisure surge cools. My insider tip: rather than fixating on Split as the connection point, price out routings through Zagreb or Frankfurt with a same-day onward flight—Croatia Airlines' short regional hop from a hub city often has significantly more last-minute space than the direct Split feeder flights during festival season, which sell out due to sailing and yachting crowds.
Because pricing on this route fluctuates heavily based on which transatlantic partner and connection city you're routed through, it's worth setting up fare alerts well before you plan to travel, so you can pounce the moment a premium seat opens up in your ideal shoulder-season window.